Description: Define inversion, which maps a nonzero extended complex number or element of the complex projective line (Riemann sphere) to its inverse. Beware of the overloading: the equality ( invc0 ) = infty is to be understood in the complex projective line, but 0 as an extended complex number does not have an inverse, which we can state as ( invc0 ) e/ CCbar . Note that this definition relies on df-bj-mulc , which does not bypass ordinary complex multiplication, but defines extended complex multiplication on top of it. Therefore, we could have used directly / instead of ( iota_ ... .cc ... ) . (Contributed by BJ, 22-Jun-2019)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-bj-invc | ⊢ -1ℂ̅ = ( 𝑥 ∈ ( ℂ̅ ∪ ℂ̂ ) ↦ if ( 𝑥 = 0 , ∞ , if ( 𝑥 ∈ ℂ , ( ℩ 𝑦 ∈ ℂ ( 𝑥 ·ℂ̅ 𝑦 ) = 1 ) , 0 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cinvc | ⊢ -1ℂ̅ | |
| 1 | vx | ⊢ 𝑥 | |
| 2 | cccbar | ⊢ ℂ̅ | |
| 3 | ccchat | ⊢ ℂ̂ | |
| 4 | 2 3 | cun | ⊢ ( ℂ̅ ∪ ℂ̂ ) |
| 5 | 1 | cv | ⊢ 𝑥 |
| 6 | cc0 | ⊢ 0 | |
| 7 | 5 6 | wceq | ⊢ 𝑥 = 0 |
| 8 | cinfty | ⊢ ∞ | |
| 9 | cc | ⊢ ℂ | |
| 10 | 5 9 | wcel | ⊢ 𝑥 ∈ ℂ |
| 11 | vy | ⊢ 𝑦 | |
| 12 | cmulc | ⊢ ·ℂ̅ | |
| 13 | 11 | cv | ⊢ 𝑦 |
| 14 | 5 13 12 | co | ⊢ ( 𝑥 ·ℂ̅ 𝑦 ) |
| 15 | c1 | ⊢ 1 | |
| 16 | 14 15 | wceq | ⊢ ( 𝑥 ·ℂ̅ 𝑦 ) = 1 |
| 17 | 16 11 9 | crio | ⊢ ( ℩ 𝑦 ∈ ℂ ( 𝑥 ·ℂ̅ 𝑦 ) = 1 ) |
| 18 | 10 17 6 | cif | ⊢ if ( 𝑥 ∈ ℂ , ( ℩ 𝑦 ∈ ℂ ( 𝑥 ·ℂ̅ 𝑦 ) = 1 ) , 0 ) |
| 19 | 7 8 18 | cif | ⊢ if ( 𝑥 = 0 , ∞ , if ( 𝑥 ∈ ℂ , ( ℩ 𝑦 ∈ ℂ ( 𝑥 ·ℂ̅ 𝑦 ) = 1 ) , 0 ) ) |
| 20 | 1 4 19 | cmpt | ⊢ ( 𝑥 ∈ ( ℂ̅ ∪ ℂ̂ ) ↦ if ( 𝑥 = 0 , ∞ , if ( 𝑥 ∈ ℂ , ( ℩ 𝑦 ∈ ℂ ( 𝑥 ·ℂ̅ 𝑦 ) = 1 ) , 0 ) ) ) |
| 21 | 0 20 | wceq | ⊢ -1ℂ̅ = ( 𝑥 ∈ ( ℂ̅ ∪ ℂ̂ ) ↦ if ( 𝑥 = 0 , ∞ , if ( 𝑥 ∈ ℂ , ( ℩ 𝑦 ∈ ℂ ( 𝑥 ·ℂ̅ 𝑦 ) = 1 ) , 0 ) ) ) |